370 
    
CT Image Line 
AP Surview 
  
  
  
  
  
  
  
CT Scan Slice System 
| Y 
+ Tete X 
x / 
LAT Focus 
+ 
CT Image Line 
LAT Surview 
AP Focus 
Figure 6 - Two dimensional central projections of the AP and LAT surviews. 
The projectivity between planes is given by equation (5) below, where all the a;; terms are divided by ap3 and new 
| 
transformation parameters hi = aj / à54 are introduced. J 
a a tata : hth thy, 
= (5) 
a,X*8,,Y+a,, hy X th, Y +1 . 
Rewriting equation (5) and gathering terms: 
X 2 h44X e hi5 Y e hi5 - ha4xX - ha5xY (6) 
There are five unknowns, the five h parameters, requiring five non-collinear points for a unique solution. If additional 
points are used for an overdetermined solution, it is possible to use the least squares solution to solve for the unknown 
h parameters. 
The least squares solution is: Ww -(A'A)'*A'l (7) 
where, for every point in matrix form: 
h 
h 
Aw=l where: w=|h, /=[x] A=[X Y 1 -xX -xY] (8) 
h 
h 
H parameters are determined for both surviews, hij and hij for the AP and LAT surviews respectively, from the x and x 
coordinates of the AP and LAT surviews respectively and the X, Y slice coordinates of the CT control points. By 
backsubstitution it is possible to solve for the X, Y slice coordinates of any additional points using the x and X 
coordinates of those points from the AP and LAT surviews respectively and the two sets of h parameters. Rewriting 
equation (6) for the AP and LAT surview: 
d m . : 9 
AP view: x=(h,4 xh, JX +(h, , xh. )Y +h, (9) 
LAT view: X=(h44-x#h214)X +(h42 - x*ho2)Y +h43 (10) 
6. ACCURACY DETERMINATION USING THE CT CONTROL AND THE PHANTOM HEAD 
The h parameters were calculated from the coordinates of the control points measured on the CT scan slice and AP 
and LAT surviews. To determine the precision of the two dimensional projective algorithm, the h parameters were used 
to calculate equivalent CT scan slice values of the control points from the AP and LAT surviews. A precision of 0.5mm 
IAPRS, Vol. 30, Part 5W1, ISPRS Intercommission Workshop “From Pixels to Sequences”, Zurich, March 22-24 1995 
  
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